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I am trying to understand some basic stochastic process simulation for population growth. Let us say, a population grows exponentially at a constant rate $R$, given by the equation, $$ P(t+1) = R P(t) $$ In the book "A Biologist's Guide to Mathematical Modeling in Ecology and Evolution" the authors generate various realizations of this model by taking a value of the rate parameter (like $R = 1.2$) and sampling from a Poisson distribution with mean $RP(t)$ to get $P(t+1)$. In reality, in some cases, we may not know the rate parameter $R$. Is there a way to estimate $R$ if we know the mean value of $P(t)$ and $P(t+1)$ at some point $t$, by such simulations? Thanks.

user2167741
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Let $p(t) = logP(t), r = log R$. Then the equation becomes $p(t+1)=r+p(t)$, and you can turn that into something like $p(t+t_0)-p(t_0)=rt$. You can run a standard linear regression on the latter.

ertl
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  • Brilliant! So you mean to say we can simulate from a Poisson for $p(t+t_0)$ and $p(t_0)$ and do a regression. But the $log$ of a Poisson will not be a Poisson right? I suppose we can still sample from Poisson and take a log. – user2167741 Aug 02 '18 at 11:35
  • Also, the above equation should be $p(t+t_0)-p(t) = rt_0$. – user2167741 Aug 02 '18 at 14:26